Existence of global weak solutions to a Cahn-Hilliard cross-diffusion system in lymphangiogenesis
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913191510933504 |
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| author | Jüngel, Ansgar Li, Yue |
| author_facet | Jüngel, Ansgar Li, Yue |
| contents | The global-in-time existence of weak solutions to a degenerate Cahn-Hilliard cross-diffusion system with singular potential in a bounded domain with no-flux boundary conditions is proved. The model consists of two coupled parabolic fourth-order partial differential equations and describes the evolution of the fiber phase volume fraction and the solute concentration, modeling the pre-patterning of lymphatic vessel morphology. The fiber phase fraction satisfies the segregation property if this holds initially. The existence proof is based on a three-level approximation scheme and a priori estimates coming from the energy and entropy inequalities. While the free energy is nonincreasing in time, the entropy is only bounded because of the cross-diffusion coupling. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_05180 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of global weak solutions to a Cahn-Hilliard cross-diffusion system in lymphangiogenesis Jüngel, Ansgar Li, Yue Analysis of PDEs 35D30, 35K35, 35K65, 35K67, 92C37 The global-in-time existence of weak solutions to a degenerate Cahn-Hilliard cross-diffusion system with singular potential in a bounded domain with no-flux boundary conditions is proved. The model consists of two coupled parabolic fourth-order partial differential equations and describes the evolution of the fiber phase volume fraction and the solute concentration, modeling the pre-patterning of lymphatic vessel morphology. The fiber phase fraction satisfies the segregation property if this holds initially. The existence proof is based on a three-level approximation scheme and a priori estimates coming from the energy and entropy inequalities. While the free energy is nonincreasing in time, the entropy is only bounded because of the cross-diffusion coupling. |
| title | Existence of global weak solutions to a Cahn-Hilliard cross-diffusion system in lymphangiogenesis |
| topic | Analysis of PDEs 35D30, 35K35, 35K65, 35K67, 92C37 |
| url | https://arxiv.org/abs/2401.05180 |